To find the general term for the sequence $\{a_n\}$ given the conditions:

$a_1 = 1$ $a_{n+1} = a_n + 3^n \quad (n = 1, 2, 3, \ldots)$

We can start by writing out the first few terms of the sequence to identify any pattern:

1. $a_1 = 1$
2. $a_2 = a_1 + 3^1 = 1 + 3 = 4$
3. $a_3 = a_2 + 3^2 = 4 + 9 = 13$
4. $a_4 = a_3 + 3^3 = 13 + 27 = 40$

Notice that each term adds the next power of 3 to the previous term. To find a pattern, we can express $a_n$ in a more general form.

Let's write $a_n$ in terms of the initial value and the sum of the powers of 3:

$a_2 = a_1 + 3^1$ $a_3 = a_1 + 3^1 + 3^2$ $a_4 = a_1 + 3^1 + 3^2 + 3^3$

In general, we can express $a_n$ as:

$a_n = a_1 + 3^1 + 3^2 + \cdots + 3^{n-1}$

The sum of the series $3^0 + 3^1 + 3^2 + \cdots + 3^{n-1}$ is a geometric series with the first term $a = 1$ (since $3^0 = 1$) and common ratio $r = 3$. The sum of the first $n$ terms of a geometric series is given by:

$S_n = a \frac{r^n - 1}{r - 1}$

For our series, $a = 1$ and $r = 3$, so:

$S_n = \frac{3^n - 1}{3 - 1} = \frac{3^n - 1}{2}$

Therefore, the general term $a_n$ can be written as:

$a_n = 1 + \frac{3^n - 3}{2}$

Simplifying further:

$a_n = 1 + \frac{3^n - 3}{2} = \frac{2 + 3^n - 3}{2} = \frac{3^n - 1}{2}$

Thus, the general term for the sequence is:

$\boxed{a_n = \frac{3^n - 1}{2}}$

Would you like more details or have any questions? Here are some questions you might want to ask next:

1. How can I prove the formula for the general term of a sequence?
2. Can you show me the derivation of the sum of a geometric series?
3. What are other common types of sequences in mathematics?
4. How can I apply this formula to find a specific term in the sequence?
5. Can you explain more about the properties of geometric sequences?

Tip: Always check the initial conditions and a few terms to verify your derived general formula.